For the case of the real billiard we shall give quite explicit formulae for the constants which express the asymptotics. We remark that, for this issue, some similar analysis has been carried out in the book [1]} (where a special attention is given to the s-called QRT maps which we do not consider here). However the present treatment is different in several respects: it is direct and essentially self-contained and develops formulae in terms of elliptic integrals which we have not found in the existing literature.

